Finite Group Databasebeta

← #14327152

Subgroups — #14327152

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1202C211yesmaximal, characteristic
2C2×#91395C212yesmaximal
3#11750768C211yesmaximal, characteristic, complemented
4C5×#4649C211yesmaximal, characteristic, complemented
5C3×#64956C211yesmaximal, characteristic, complemented
6#11748894C212yesmaximal, complemented
7#11748891C212yesmaximal, complemented
8#11748888C211yesmaximal, characteristic, complemented
9D5×C3⋊#95C211yesmaximal, characteristic, complemented
10#11748820C212yesmaximal, complemented
11#11748600C211yesmaximal, characteristic, complemented
12#10605501—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C22×#1202C2211yescharacteristic
16C22×#1200C2212yes—
17C22×D5×Dic3C2212yes—
18C2×#6714C2212yes—
19C2×#6712C2212yes—
20C2×C5×Dic3⋊C4C2212yes—
21C2×C4×C5×Dic3C2212yes—
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C3×#685C2212yes—
24C2×C3×#683C2212yes—
25C2×Dic3⋊Dic5C2212yes—
26C2×Dic3⋊Dic5C2211yescharacteristic
27C2×#6667C2211yescharacteristic
28C2×#6665C2211yescharacteristic
29C4×#1202C418yescomplemented
30#91395—28no—
31C3×C5×#95C2211yescharacteristic
32#91115C2211yescharacteristic
33C5×C3⋊#95C2211yescharacteristic
34C3×#3963C2211yescharacteristic
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#74011—51nomaximal, Hall
39C2×C4×D20—32no—
40C2×C4×D20—31no—
41C4×C5×C4⋊C4—31no—
42#65278—32no—
43#65278—32no—
44#65225—31no—
45#65217—32no—
46#64956—31no—
47#64956D311yescharacteristic, complemented
48D5×#95—31no—
49D5×#95—31no—
50C23×D15C2311yescharacteristic
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C2×C4×D15—42no—
57C2×C4×D15C2×C414yescomplemented
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2311yescharacteristic
63C2×C3×C4×D5—42no—
64C2×C3×C4×D5C2×C414yescomplemented
65C2×#1202C2×C414yes—
66C2×#1202D414yes—
67C2×#1202Q814yes—
68C2×#1200—24no—
69C2×#1200—24no—
70C2×#1200—42no—
71C2×D5×Dic3—24no—
72C2×D5×Dic3—24no—
73C2×D5×Dic3—42no—
74#6714—24no—
75#6712—42no—
76C5×Dic3⋊C4—24no—
77C4×C5×Dic3—24no—
78C4×C5×Dic3C2×C414yescomplemented
79C3×#685—24no—
80C3×#683—42no—
81Dic3⋊Dic5—24no—
82Dic3⋊Dic5C2×C414yescomplemented
83#6667C2×C414yescomplemented
84#6665C2×C414yescomplemented
85C2×C4×#67—51no—
86C2×#67⋊C4—52no—
87C3×#380—51no—
88C2×C3⋊#95—51no—
89#4989—52no—
90#4976—52no—
91#4946—52no—
92#4667—51no—
93D3×#95—51no—
94#4650—51no—
95#4649D511yescharacteristic, complemented
96C22×D20—31no—
97C22×C4×D5—31no—
98C22×C4×D5—32no—
99C22×C4×D5—32no—
100C22×C4×D5C2×D311yescharacteristic
101C2×C5×C4⋊C4—32no—
102C2×C42×C5—31no—
103C2×C42×C5—32no—
104C2×#685—31no—
105C2×#685—31no—
106C2×#685—32no—
107C2×#685C2×D312yes—
108C2×C4⋊Dic5—31no—
109C2×C4⋊Dic5—32no—
110C2×#683—32no—
111C2×#683C2×D312yes—
112C4×D20—38no—
113C4×D20—68no—
114C5×#95—31no—
115C5×#95C2×D311yescharacteristic
116#3963—31no—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#3963C2×D311yescharacteristic
121#2349—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×D15—22no—
124C22×D15—22no—
125C22×D15C22×C412yes—
126C22×D15C2×D412yes—
127C22×D15C2×Q812yes—
128C22×C3×D5—22no—
129C22×C3×D5—22no—
130C22×C3×D5C22×C412yes—
131C22×C3×D5C2×D412yes—
132C22×C3×D5C2×Q812yes—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5C22×C412yes—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15C22×C412yes—
143C4×D15—44no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3C22×C412yes—
148C2×C5×Dic3C22×C412yes—
149C2×C5×Dic3C2×D412yes—
150C2×C5×Dic3C2×Q812yes—
151C2×C5×Dic3#4114yes—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5C22×C412yes—
157C3×C4×D5—44no—
158#1202C4⋊C4116yescomplemented
159#1200—48no—
160D5×Dic3—48no—
161C23×C3×C4—51no—
162C22×#67—51no—
163C23×Dic3—52no—
164C22×C4×D3—51no—
165C22×C4×D3—52no—
166C2×C3×C4⋊C4—52no—
167C2×C3×C22⋊C4—52no—
168C2×#217—51no—
169C4×#67—58no—
170C2×(C2×D3)⋊C4—51no—
171C2×(C2×D3)⋊C4—52no—
172C2×Dic3⋊C4—51no—
173C2×Dic3⋊C4—52no—
174C2×Dic3⋊C4—52no—
175C2×Dic3⋊C4C2×D512yes—
176C2×C4×Dic3C2×D512yes—
177C2×C4×Dic3C2×D511yescharacteristic
178#67⋊C4—108no—
179C3×#95—51no—
180C3×#95C2×D511yescharacteristic
181C3⋊#95—51no—
182C3⋊#95—52no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95C2×D511yescharacteristic
186C23×D5—31no—
187C23×D5C22×D311yescharacteristic
188C22×C4×C5—31no—
189C22×C4×C5—31no—
190C22×C4×C5—32no—
191C22×C4×C5C22×D312yes—
192C22×C4×C5C22×D311yescharacteristic
193C22×Dic5—31no—
194C22×Dic5—32no—
195C22×Dic5C22×D312yes—
196C22×Dic5C22×D311yescharacteristic
197C2×D20—34no—
198C2×D20—34no—
199C2×D20—34no—
200C2×C4×D5—34no—
201C2×C4×D5—42no—
202C2×C4×D5—64no—
203C2×C4×D5—64no—
204C2×C4×D5—64no—
205C2×C4×D5—64no—
206C2×C4×D5—122no—
207C2×C4×D5—122no—
208C2×C4×D5—122no—
209C2×C4×D5C4×D314yescomplemented
210C5×C4⋊C4—64no—
211C42×C5—34no—
212C42×C5—64no—
213#685—24no—
214#685—34no—
215#685—34no—
216#685—64no—
217C4⋊Dic5—34no—
218C4⋊Dic5—64no—
219#683—42no—
220#683—122no—
221C2×C4×D4—152no—
222C2×C4×D4—151no—
223#389—152no—
224#385—152no—
225#385—152no—
226#382—151no—
227#380—51no—
228#380—151no—
229C4×C4⋊C4—31no—
230C2×#95—151no—
231C2×#95—151no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C22×Q811yescharacteristic
235C22×C3×C5C2×#4112yes—
236C22×C3×C5H(5,𝔽2)12yes—
237C2×D15—22no—
238C2×D15—22no—
239C2×D15—24no—
240C2×D15—24no—
241C2×D15C2×C4⋊C414yes—
242C2×C3×D5—22no—
243C2×C3×D5—22no—
244C2×C3×D5—24no—
245C2×C3×D5—24no—
246C2×C3×D5C2×C4⋊C414yes—
247C3×C4×C5—22no—
248C3×C4×C5—42no—
249Dic15—22no—
250Dic15—42no—
251C3×Dic5—22no—
252C3×Dic5—42no—
253C5×Dic3—22no—
254C5×Dic3—24no—
255C5×Dic3C2×C4⋊C414yes—
256C24×C3—51no—
257C23×D3—51no—
258C22×C3×C4—51no—
259C22×C3×C4—54no—
260C22×C3×C4—52no—
261C22×C3×C4—202no—
262C22×C3×C4C22×D512yes—
263C22×C3×C4C22×D511yescharacteristic
264C2×#67—54no—
265C2×#67—54no—
266C2×#67—54no—
267C22×Dic3—51no—
268C22×Dic3—52no—
269C22×Dic3—104no—
270C22×Dic3—104no—
271C22×Dic3—202no—
272C22×Dic3C22×D511yescharacteristic
273C22×Dic3C22×D512yes—
274C22×Dic3C22×D511yescharacteristic
275C2×C4×D3—54no—
276C2×C4×D3—104no—
277C2×C4×D3—104no—
278C2×C4×D3—202no—
279C2×C4×D3—202no—
280C3×C4⋊C4—202no—
281C3×C22⋊C4—104no—
282#217—54no—
283(C2×D3)⋊C4—54no—
284(C2×D3)⋊C4—104no—
285Dic3⋊C4—24no—
286Dic3⋊C4—54no—
287Dic3⋊C4—104no—
288Dic3⋊C4—202no—
289C4×Dic3—24no—
290C4×Dic3C4×D514yescomplemented
291C23×C5C23×D311yescharacteristic
292C22×D5—22no—
293C22×D5—22no—
294C22×D5—32no—
295C22×D5—32no—
296C22×D5—32no—
297C22×D5—62no—
298C22×D5—62no—
299C22×D5C2×C4×D312yes—
300C22×D5D3×D412yes—
301C22×D5D3×Q812yes—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—32no—
307C2×C4×C5—32no—
308C2×C4×C5—32no—
309C2×C4×C5—32no—
310C2×C4×C5—34no—
311C2×C4×C5—62no—
312C2×C4×C5—62no—
313C2×C4×C5—62no—
314C2×C4×C5C2×C4×D312yes—
315C2×Dic5—22no—
316C2×Dic5—22no—
317C2×Dic5—22no—
318C2×Dic5—22no—
319C2×Dic5—32no—
320C2×Dic5—62no—
321C2×Dic5—62no—
322C2×Dic5—62no—
323C2×Dic5—62no—
324C2×Dic5C2×C4×D312yes—
325D20—316no—
326C4×D5—44no—
327C4×D5—128no—
328C4×D5—124no—
329C4×D5—128no—
330C22×D4—151no—
331C23×C4—51no—
332C23×C4—152no—
333C23×C4—152no—
334C23×C4—151no—
335C4×D4—158no—
336C4×D4—308no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—52no—
339C2×C4⋊C4—151no—
340C2×C4⋊C4—152no—
341C2×C4⋊C4—152no—
342C2×C22⋊C4—52no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—151no—
345C2×C22⋊C4—152no—
346C2×C42—31no—
347C2×C42—32no—
348#95—31no—
349#95—51no—
350#95—152no—
351#95—152no—
352#95—152no—
353#95—151no—
354#95D3×D511yescharacteristic
355C2×C3×C5C22×C4⋊C411yescharacteristic
356C2×C3×C5C4×#4111yescharacteristic
357C2×C3×C5#51811yescharacteristic
358C2×C3×C5#54612yes—
359C2×C3×C5#55412yes—
360D15—24no—
361C3×D5—24no—
362C23×C3—52no—
363C23×C3—52no—
364C23×C3—52no—
365C23×C3—102no—
366C23×C3—102no—
367C23×C3C23×D511yescharacteristic
368C22×D3—52no—
369C22×D3—52no—
370C22×D3—52no—
371C22×D3—102no—
372C22×D3—102no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—52no—
378C2×C3×C4—102no—
379C2×C3×C4—102no—
380C2×C3×C4—102no—
381C2×C3×C4—102no—
382C2×C3×C4—204no—
383C2×C3×C4C2×C4×D512yes—
384#67—516no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—52no—
389C2×Dic3—102no—
390C2×Dic3—102no—
391C2×Dic3—102no—
392C2×Dic3—102no—
393C2×Dic3—208no—
394C2×Dic3C2×C4×D512yes—
395C2×Dic3C2×C4×D512yes—
396C2×Dic3#70914yes—
397C2×Dic3D4×D512yes—
398C2×Dic3Q8×D512yes—
399C4×D3—208no—
400C4×D3—204no—
401C22×C5C22×C4×D311yescharacteristic
402C22×C5C2×D3×D411yescharacteristic
403C22×C5#100412yes—
404C22×C5C2×D3×Q811yescharacteristic
405C22×C5D3×#4112yes—
406C2×D5—24no—
407C2×D5—22no—
408C2×D5—22no—
409C2×D5—24no—
410C2×D5—34no—
411C2×D5—64no—
412C2×D5—62no—
413C2×D5—62no—
414C2×D5—64no—
415C2×D5D3×C4⋊C414yes—
416C4×C5—22no—
417C4×C5—34no—
418C4×C5—42no—
419C4×C5—62no—
420C4×C5—64no—
421Dic5—22no—
422Dic5—42no—
423Dic5—62no—
424Dic5—122no—
425C24—51no—
426C24—151no—
427C2×D4—154no—
428C2×D4—154no—
429C2×D4—154no—
430C22×C4—31no—
431C22×C4—32no—
432C22×C4—31no—
433C22×C4—51no—
434C22×C4—54no—
435C22×C4—52no—
436C22×C4—151no—
437C22×C4—154no—
438C22×C4—152no—
439C22×C4—202no—
440C22×C4—304no—
441C22×C4—304no—
442C22×C4—304no—
443C22×C4—304no—
444C22×C4—602no—
445C22×C4—602no—
446C22×C4—602no—
447C22×C4C2×D3×D512yes—
448C22×C4C2×D3×D511yescharacteristic
449C4⋊C4—64no—
450C4⋊C4—154no—
451C4⋊C4—202no—
452C4⋊C4—304no—
453C4⋊C4—602no—
454C22⋊C4—104no—
455C22⋊C4—154no—
456C22⋊C4—154no—
457C22⋊C4—304no—
458C42—34no—
459C42—64no—
460C3×C5#234911yescharacteristic, complemented, Hall
461C22×C3—54no—
462C22×C3—102no—
463C22×C3—102no—
464C22×C3—104no—
465C22×C3—104no—
466C22×C3C22×C4×D511yescharacteristic
467C22×C3C2×#70912yes—
468C22×C3C2×D4×D511yescharacteristic
469C22×C3C2×Q8×D511yescharacteristic
470C22×C3#414912yes—
471C2×D3—54no—
472C2×D3—102no—
473C2×D3—102no—
474C2×D3—104no—
475C2×D3—104no—
476C3×C4—22no—
477C3×C4—42no—
478C3×C4—102no—
479C3×C4—202no—
480Dic3—22no—
481Dic3—24no—
482Dic3—102no—
483Dic3—202no—
484Dic3D5×C4⋊C414yes—
485C2×C5#548611yescharacteristic
486C2×C5C2×D3×C4⋊C411yescharacteristic
487C2×C5#552211yescharacteristic
488C2×C5#561012yes—
489C2×C5#563612yes—
490D5—24no—
491D5—64no—
492C23—52no—
493C23—52no—
494C23—52no—
495C23—102no—
496C23—102no—
497C23—152no—
498C23—152no—
499C23—152no—
500C23—302no—
501C23—302no—
502C23C22×D3×D511yescharacteristic
503D4—1516no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—32no—
511C2×C4—32no—
512C2×C4—34no—
513C2×C4—52no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—102no—
518C2×C4—102no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—152no—
522C2×C4—204no—
523C2×C4—302no—
524C2×C4—302no—
525C2×C4—302no—
526C2×C4—302no—
527C2×C4—604no—
528C2×C4—608no—
529C2×C4—608no—
530C2×C4C4×D3×D512yes—
531C2×C3—104no—
532C2×C3C4×#70911yescharacteristic
533C2×C3C2×D5×C4⋊C411yescharacteristic
534C2×C3#6585511yescharacteristic
535C2×C3#6588412yes—
536C2×C3#6591312yes—
537D3—104no—
538C5#7401111yescharacteristic, complemented, Sylow
539C22—54no—
540C22—104no—
541C22—102no—
542C22—102no—
543C22—104no—
544C22—154no—
545C22—304no—
546C22—302no—
547C22—302no—
548C22—304no—
549C22C2×C4×D3×D511yescharacteristic
550C22D3×#70912yes—
551C22D3×D4×D511yescharacteristic
552C22#9202612yes—
553C22D3×Q8×D511yescharacteristic
554C4—22no—
555C4—34no—
556C4—42no—
557C4—64no—
558C4—62no—
559C4—102no—
560C4—202no—
561C4—302no—
562C4—602no—
563C3#1060550111yescharacteristic, complemented, Sylow
564C2—104no—
565C2—304no—
566C2#1175266611yescharacteristic
567C2#1175306611yescharacteristic
568C2#1175324112yes—
569C2#1175345112yes—
570C2D3×D5×C4⋊C411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1202C22
hypercenter#95D3×D5
hypocenterC3×C5#2349
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11750768C2semidirect
4. C5×#4649C2semidirect
5. C3×#64956C2semidirect
6. #11748894C2semidirect
7. #11748891C2semidirect
8. #11748888C2semidirect
9. D5×C3⋊#95C2semidirect
10. #11748820C2semidirect
11. #11748600C2semidirect
29. C4×#1202C4semidirect
47. #64956D3semidirect
57. C2×C4×D15C2×C4semidirect
64. C2×C3×C4×D5C2×C4semidirect
78. C4×C5×Dic3C2×C4semidirect
82. Dic3⋊Dic5C2×C4semidirect
83. #6667C2×C4semidirect
84. #6665C2×C4semidirect
95. #4649D5semidirect
158. #1202C4⋊C4semidirect
209. C2×C4×D5C4×D3semidirect
290. C4×Dic3C4×D5semidirect
460. C3×C5#2349semidirect
538. C5#74011semidirect
563. C3#10605501semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×#12022
C2×#913952
#117507682
C5×#46492
C3×#649562
#117488942
#117488912
#117488882
D5×C3⋊#952
#117488202
#117486002
#106055013
#740115